Problem 54 - College Algebra

Question

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$ f(x)=x(14-2 x)(10-2 x) $$

Step-by-Step Solution

Answer

Intercepts: (0,0), (5,0), (7,0); End behavior: \( x \to \infty, f(x) \to -\infty \), \( x \to -\infty, f(x) \to \infty \).

Step 1: Understand the Polynomial Structure

The given polynomial function is \( f(x) = x(14-2x)(10-2x) \). This expression is already factored, making it easier to identify the x-intercepts, which occur where the function equals zero.

Step 2: Find the x-intercepts

Set each factor equal to zero to find the x-intercepts:- \( x = 0 \)- \( 14 - 2x = 0 \) leads to \( x = 7 \)- \( 10 - 2x = 0 \) leads to \( x = 5 \)Thus, the x-intercepts are \( x = 0 \), \( x = 5 \), and \( x = 7 \).

Step 3: Find the y-intercept

To find the y-intercept, substitute \( x = 0 \) into the function: \[ f(0) = 0(14 - 0)(10 - 0) = 0 \]Therefore, the y-intercept is \( (0, 0) \).

Step 4: Determine the End Behavior

The leading term of the expanded polynomial determines the end behavior. The highest degree term can be found by multiplying the highest degree terms from each factor: \( x \) from \( x \), \( -2x \) from \( 14 - 2x \), \( -2x \) from \( 10 - 2x \)This gives \( -4x^3 \). Since the leading coefficient is negative and the degree is odd, as \( x \to \infty \), \( f(x) \to -\infty \), and as \( x \to -\infty \), \( f(x) \to \infty \).

Step 5: Graph the Polynomial

Using a calculator, graph the polynomial function to visualize the intercepts and end behavior. The graph should show:- Intercepts at \( (0,0) \), \( (5,0) \), and \( (7,0) \)- As \( x \to \infty \), the graph goes downwards, and as \( x \to -\infty \), it goes upwards.

Key Concepts

Graphing Polynomial Functions

Graphing polynomial functions might seem daunting, but it becomes more approachable with a step-by-step approach. The function we are dealing with is a third-degree polynomial of the form \( f(x) = x(14-2x)(10-2x) \). Start by understanding its nature — each term's degree gives us a clue about how the graph might look. Third-degree polynomials can have up to three x-intercepts and will generally exhibit a wiggle or curve characteristic of polynomials of odd degree.
To graph this function:

• Identify the x-intercepts where the function crosses the x-axis.
• Determine the y-intercept where the graph crosses the y-axis.
• Evaluate the end behavior to know the direction of the ends of the graph.

Making use of graphing calculators can simplify this process, enabling the visualization of the polynomial’s shape, and highlighting key features like intercepts and behavior at each end.

Intercepts

Finding intercepts in polynomial functions is akin to uncovering their zero points. The x-intercepts are found where the function \( f(x) \) equals zero. For our function \( x(14-2x)(10-2x) \), this happens at the values of \( x \) which make any factor zero:

• \( x = 0 \)
• \( 14 - 2x = 0 \) resolves to \( x = 7 \)
• \( 10 - 2x = 0 \) resolves to \( x = 5 \)

Thus, the x-intercepts are \( x = 0 \), \( x = 5 \), and \( x = 7 \), corresponding to the points where the graph will cross the x-axis. The y-intercept occurs where the graph crosses the y-axis, which is simply found by evaluating \( f(0) \). Substituting zero gives us \( f(0) = 0 \) indicating the point \((0,0)\) is both an x-intercept and the y-intercept.

End Behavior

Understanding end behavior in a polynomial function involves analyzing the term with the highest degree. For the polynomial \( x(14-2x)(10-2x) \), upon expanding, the lead term is \( -4x^3 \). This is a third-degree term which crucially hints at the graph's endpoints direction:

• A negative leading coefficient means the graph dips downwards to the right, as \( x \rightarrow \infty \), \( f(x) \rightarrow -\infty \).
• Conversely, the graph climbs upwards to the left as \( x \rightarrow -\infty \), \( f(x) \rightarrow \infty \).

The odd degree implies the ends of the graph diverge in opposite directions - one upwards, the other downwards. This behavior can be visually assessed by sketching or using digital graphing tools.

Factored Form

The factored form of a polynomial function is a crucial mathematical expression that simplifies the process of identifying roots and intercepts. In the case of the polynomial \( x(14-2x)(10-2x) \), it is already in its factored form. Each component or factor of this expression directly reveals its x-intercepts:

• \( x = 0 \)
• \( 14-2x=0 \) gives \( x = 7 \)
• \( 10-2x=0 \) gives \( x = 5 \)

This form allows us to immediately identify where the function becomes zero, showcasing the graph's touch or crossing points on the x-axis. Additionally, factored form aids in recognizing repeated roots and complicating polynomial behaviors like turning points. It’s advantageous for analyzing more intricate features of polynomial graphs.